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NAG Toolbox: nag_quad_1d_fin_well (d01ah)

Purpose

nag_quad_1d_fin_well (d01ah) computes a definite integral over a finite range to a specified relative accuracy using a method described by Patterson.

Syntax

[result, npts, relerr, ifail] = d01ah(a, b, epsr, f, nlimit)
[result, npts, relerr, ifail] = nag_quad_1d_fin_well(a, b, epsr, f, nlimit)

Description

nag_quad_1d_fin_well (d01ah) computes a definite integral of the form
b
f(x)dx.
a
abf(x)dx.
The method uses as its basis a family of interlacing high precision rules (see Patterson (1968)) using 11, 33, 77, 1515, 3131, 6363, 127127 and 255255 nodes. Initially the family is applied in sequence to the integrand. When two successive rules differ relatively by less than the required relative accuracy, the last rule used is taken as the value of the integral and the operation is regarded as successful. If all rules in the family have been applied unsuccessfully, subdivision is invoked. The subdivision strategy is as follows. The interval under scrutiny is divided into two sub-intervals (not always equal). The basic family is then applied to the first sub-interval. If the required accuracy is not obtained, the interval is stored for future examination (see ifail = 2ifail=2) and the second sub-interval is examined. Should the basic family again be unsuccessful, then the sub-interval is further subdivided and the whole process repeated. Successful integrations are accumulated as the partial value of the integral. When all possible successful integrations have been completed, those previously unsuccessful sub-intervals placed in store are examined.
A large number of refinements are incorporated to improve the performance. Some of these are:
(a) The rate of convergence of the basic family is monitored and used to make a decision to abort and subdivide before the full sequence has been applied.
(b) The εε-algorithm is applied to the basic results in an attempt to increase the convergence rate. See Wynn (1956).
(c) An attempt is made to detect sharp end point peaks and singularities in each sub-interval and to apply appropriate transformations to smooth the integrand. This consideration is also used to select interval sizes in the subdivision process.
(d) The relative accuracy sought in each sub-interval is adjusted in accordance with its likely contribution to the total integral.
(e) Random transformations of the integrand are applied to improve reliability in some instances.

References

Patterson T N L (1968) The Optimum addition of points to quadrature formulae Math. Comput. 22 847–856
Wynn P (1956) On a device for computing the em(Sn)em(Sn) transformation Math. Tables Aids Comput. 10 91–96

Parameters

Compulsory Input Parameters

1:     a – double scalar
aa, the lower limit of integration.
2:     b – double scalar
bb, the upper limit of integration. It is not necessary that a < ba<b.
3:     epsr – double scalar
The relative accuracy required.
Constraint: epsr > 0.0epsr>0.0.
4:     f – function handle or string containing name of m-file
f must return the value of the integrand ff at a given point.
[result] = f(x)

Input Parameters

1:     x – double scalar
The point at which the integrand ff must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
5:     nlimit – int64int32nag_int scalar
A limit to the number of function evaluations. If nlimit0nlimit0, the function uses a default limit of 1000010000.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     npts – int64int32nag_int scalar
The number of function evaluations used in the calculation of the integral.
3:     relerr – double scalar
A rough estimate of the relative error achieved.
4:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Note: nag_quad_1d_fin_well (d01ah) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1ifail=1
The integral has not converged to the accuracy requested. It may be worthwhile to try increasing nlimit.
W ifail = 2ifail=2
Too many unsuccessful levels of subdivision have been invoked.
  ifail = 3ifail=3
On entry,epsr0.0epsr0.0.
When ifail = 1ifail=1 or 22 a result may be obtained by continuing without further subdivision, but this is likely to be inaccurate.

Accuracy

The relative accuracy required is specified by you in the variable epsr. The function will terminate whenever the relative accuracy specified by epsr is judged to have been reached.
If on exit, ifail = 0ifail=0, then it is most likely that the result is correct to the specified accuracy. If, on exit, ifail = 1ifail=1 or 22, then it is likely that the specified accuracy has not been reached.
relerr is a rough estimate of the relative error achieved. It is a by-product of the computation and is not used to effect the termination of the function. The outcome of the integration must be judged by the value of ifail.

Further Comments

The time taken by nag_quad_1d_fin_well (d01ah) depends on the complexity of the integrand and the accuracy required.

Example

function nag_quad_1d_fin_well_example
a = 0;
b = 1;
epsr = 1e-05;
nlimit = int64(0);
f = @(x) 4.0/(1.0+x^2);
[result, npts, relerr, ifail] = nag_quad_1d_fin_well(a, b, epsr, f, nlimit)
 

result =

    3.1416


npts =

                   15


relerr =

   5.8494e-09


ifail =

                    0


function d01ah_example
a = 0;
b = 1;
epsr = 1e-05;
nlimit = int64(0);
f = @(x) 4.0/(1.0+x^2);
[result, npts, relerr, ifail] = d01ah(a, b, epsr, f, nlimit)
 

result =

    3.1416


npts =

                   15


relerr =

   5.8494e-09


ifail =

                    0



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Chapter Introduction
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